Sound (things to remember)

What is sound?

Sound is a form of wave in which a pattern of pressure is propagated through an elastic medium. A sound can propagate only in a medium (or matter) which has mass and elasticity, hence it cannot travel in vacuum as it is essentially empty of matter.

Sound can travel in any medium or matter such as liquids, solids, gases and plasma. The sound propagates in a medium such as air, as a series of compressions and rarefactions in the density of particles of air (also known as longitudinal waves). A compression is a positive pressure change and a rarefaction is a negative pressure change relative to the static atmospheric pressure.

Sound is generated when a body is set into vibration. The vibratory motion consists of back and forth movement of the body that has the properties of mass and elasticity. The back and forth movement occurs as a result of two opposing forces, inertia and elasticity. Mass and elasticity are the two critical properties for the source of sound and the medium through which sound is propagated.

How mass and elasticity are critical in sound generation?

Consider a tuning fork, it has mass and it also possess the restoring force of elasticity. When the tuning fork is struck gently, the tines are set in to vibration, this takes the form of back and forth movements. Newton’s law of motion states that every object will remain at rest or in uniform motion in a straight line unless forced to change its state by the action of another force. This property of the object to remain in a state of rest or in uniform motion is called as inertia and the amount of inertia is directly proportional to the mass of the object.

When the tuning fork is struck gently, the tine moves in the direction of external force due to inertia. Now the elasticity causes the tines to move in another direction to return to equilibrium, and an inertial force causes the tines to move through equilibrium toward maximum displacement in the opposite direction. The process continues as the elasticity returns the tines back toward equilibrium and the force of inertia causes the tines to be displaced beyond equilibrium and so on. Because of the interaction of the two forces, elasticity and inertia, the tines continue to move in back and forth motion and the fork is set into vibration, and sound is generated.

Ratios, Logarithms, and Decibels: Laws of logarithms (things to remember)

This post provides the laws of logarithms that may be frequently used to solve problems involving logarithms.

Law 1: log (m*n) = log (m) + log (n)

Example:
log (2*3) = log (2) + log (3);
log (2*3) = 0.3010 + 0.4771;
log (2*3) = 0.7781;

Law 2: log (m/n) = log (m) - log (n)

Example:
log (5/3) = log (5) - log (3);
log (5/3) = 0.6989 - 0.4771;
log (5/3) = 0.2218;

Law 3: log m^n = n * log (m)

Example:
log (5^2) = 2 * log (5);
log (5^2) = 2 * 0.6989;
log (5^2) = 1.3978;

Law 4: log (1/m) = - log (m)

Example:
log (1/2) = - log (2);
log (1/2) = - 0.3010;

Ratios, Logarithms, and Decibels (things to remember)

When appearing for competitive examinations or elsewhere, it is important to find solutions quickly and near accurately to mathematical problem involving logarithms (especially when Log Table is not available). In such situation, the problem involving logarithm can be easily and quickly solved almost accurately by remembering the logs of 1, 2, 3, 7 and 10 with the knowledge of ‘Laws of Logarithms’.

log of 1 = 0
log of 2 = 0.3
log of 3 = 0.48
log of 7 = 0.85
log of 10 = 1

Examples

1. log of 4 = ?

We know from the Laws of logarithm that,

log (m*n) = log (m) + log (n)

Since we know the log of 2 is 0.3 and 4 is a multiple of 2. In our example, 4 can be rewritten as 2 multiplied by 2 i.e., 4 = 2*2. Hence, log 4 can be rewritten as log (2*2).

log (4) = log (2*2) = log (2) + log (2);
log (4) = 0.3 + 0.3;
log (4) = 0.6 ; From log table, log (4) = 0.6020

2. log of 5 = ?

We know from the Laws of logarithm that,

log (m/n) = log (m) - log (n)

In this example, 5 can be rewritten as 10 divided by 2 i.e., 5 = 10/2. Hence, log (5) can be rewritten as log (10/2).

log (5) = log (10/2) = log (10) – log (2);
log (5) = 1 – 0.3;

log (5) = 0.7 ; From log table, log (5) = 0.6989